Systems and methods for avian flock flight path modification using UAVs

ABSTRACT

Systems and methods for autonomously herding birds in accordance with embodiments of the invention are illustrated. One embodiment includes an autonomous flock herding system, including a bird location sensor, a drone; and a control system, including a processor, and a memory, the memory containing a flock herding application, where the application directs the processor to obtain bird position data from the at least one bird location sensor, where the bird position data describes the location of birds in a flock of birds, determine if the flock of birds will enter a protected zone, generate a set of waypoints using a flock dynamics model, instruct the unmanned aerial vehicle to navigate to at least one waypoint in the set of waypoints such that the flock of birds will, in response to the presence of the unmanned aerial vehicle at the at least one waypoint, change trajectory away from the protected zone.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present invention claims priority to U.S. Provisional PatentApplication Ser. No. 62/703,759 entitled “Robotic Herding of a Flock ofBirds using an Unmanned Aerial Vehicle” to Chung et al., filed Jul. 26,2018, the disclosure of which is herein incorporated by reference in itsentirety.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under Grant No(s).IIS1253758 and IIS1664186 awarded by the National Science Foundation.The government has certain rights in the invention.

FIELD OF THE INVENTION

The present invention generally relates to the herding of avians usingunmanned aerial vehicles (UAVs, or colloquially “Drones”), and namelysystems and methods for modifying flight path while maintaining flockintegrity.

BACKGROUND

Unmanned Aerial Vehicles (UAVs), also referred to as “Drones” areautonomous robotic vehicles capable of flight. One of the most commonUAV designs is the quadcopter, a four-rotor device capable of immediatemovement in each of the three axes. However, there are many fixed-wingUAV designs as well.

Sharing the skies, many avian species travel in flocks. Flocks can bemade up of any number of birds, from two to tens of thousands, dependingon the species. Mixed flocks can also form from birds of two or morespecies. Flock movements, such as their flight paths and formations, arehighly complex and dependent upon many factors.

SUMMARY OF THE INVENTION

Systems and methods for autonomously herding birds in accordance withembodiments of the invention are illustrated. One embodiment includes anautonomous bird flock herding system, including at least one birdlocation sensor, an unmanned aerial vehicle; and a flock herding controlsystem, including a processor, and a memory, the memory containing aflock herding application, where the flock herding application directsthe processor to obtain bird position data from the at least one birdlocation sensor, where the bird position data describes the location ofbirds in a flock of birds, determine if the flock of birds will enter aprotected zone, generate a set of waypoints using a flock dynamicsmodel, instruct the unmanned aerial vehicle to navigate to at least onewaypoint in the set of waypoints such that the flock of birds will, inresponse to the presence of the unnamed aerial vehicle at the at leastone waypoint, change trajectory away from the protected zone.

In another embodiment, in response to the presence of the unmannedaerial vehicle, the flock maintains integrity.

In a further embodiment, the protected zone is a cylinder.

In still another embodiment, the protected zone is over an airport.

In a still further embodiment, the at least one bird location sensor isan avian radar.

In yet another embodiment the system includes at least one environmentalsensor, and the flock herding application further directs the processorto obtain environment data describing environmental conditions proximalto the protected zone from the at least one environmental sensor, andgenerate the set of waypoints using the environment data.

In a yet further embodiment, the environment data describes the positionof airplanes.

In another additional embodiment, the environment data describes windspeed.

In a further additional embodiment, the flock herding applicationfurther directs the processor to obtain updated bird position data, andupdate the set of waypoints based on the updated bird position data.

In another embodiment again, the system includes a second unmannedaerial vehicle, and wherein the flock herding application furtherdirects the processor to generate a second set of waypoints using theflock dynamics model, and instruct the second unmanned aerial vehicle tonavigate to a given at least one waypoint in the second set of waypointssuch that the flock of birds will, in response to the presence of thesecond unmanned aerial vehicle at the given at least one waypoint,change trajectory away from the protected zone.

In still yet another embodiment, in response to the presence of theunmanned aerial vehicle, the flock maintains integrity.

In a still yet further embodiment, the protected zone is a cylinder.

In still another additional embodiment, the protected zone is over anairport.

In a still further additional embodiment, the at least one bird locationsensor is an avian radar.

In still another embodiment again, the method further includesobtaining, using the flock herding control system, environment data fromat least one environmental sensor describing environmental conditionsproximal to the protected zone from the at least one environmentalsensor, and generating the set of waypoints using the environment data.

In a still further embodiment again, the environment data describes theposition of airplanes.

In yet another additional embodiment, the environment data describeswind speed.

In a yet further additional embodiment, the method further includesobtaining, using the flock herding control system, updated bird positiondata, and updating, using the flock herding control system, the set ofwaypoints based on the updated bird position data.

In yet another embodiment again, the method further includes generating,using the flock herding control system, a second set of waypoints usingthe flock dynamics model, and instructing, using the flock herdingcontrol system, a second unmanned aerial vehicle to navigate to a givenat least one waypoint in the second set of waypoints such that the flockof birds will, in response to the presence of the second unmanned aerialvehicle at the given at least one waypoint, change trajectory away fromthe protected zone.

Additional embodiments and features are set forth in part in thedescription that follows, and in part will become apparent to thoseskilled in the art upon examination of the specification or may belearned by the practice of the invention. A further understanding of thenature and advantages of the present invention may be realized byreference to the remaining portions of the specification and thedrawings, which forms a part of this disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The description and claims will be more fully understood with referenceto the following figures and data graphs, which are presented asexemplary embodiments of the invention and should not be construed as acomplete recitation of the scope of the invention.

FIG. 1 is a conceptual system diagram for a flock herding system inaccordance with an embodiment of the invention.

FIG. 2 is a conceptual block diagram for a flock herding control systemin accordance with an embodiment of the invention.

FIG. 3 is a high-level illustration of a flock herding system in actionin accordance with an embodiment of the invention.

FIG. 4 is a flowchart illustrating a flock herding process in accordancewith an embodiment of the invention.

FIG. 5 illustrates several important distance variables in the flockingmodel in accordance with an embodiment of the invention.

FIG. 6 illustrates a top view of two loci of candidate target points inaccordance with an embodiment of the invention.

FIG. 7 illustrates a typical subgraph in a dense flock and a subgraphwith two birds collinear with a pursuer in accordance with an embodimentof the invention.

FIG. 8 illustrates a model with a single edge and the pursuer inaccordance with an embodiment of the invention.

FIG. 9 illustrates two canonical scenarios with different signs of q_(k)in accordance with an embodiment of the invention.

FIG. 10 is psuedocode for the m-waypoint herding algorithm in accordancewith an embodiment of the invention.

FIG. 11 is psuedocode for the selection of m-waypoints in accordancewith an embodiment of the invention.

FIG. 12 illustrates one half of the cone outside which the herdingshould start in accordance with an embodiment of the invention.

FIG. 13 is a vector diagram illustrating push from the aft and from thefront, respectively.

FIG. 14 is an example of flock motion in the presence of a pursuer andwithout a pursuer in accordance with an embodiment of the invention.

DETAILED DESCRIPTION

Turning now to the drawings, systems and methods for avian flight pathmodification using UAVs are illustrated. Flocks of birds are a commonsight, and their dynamic movements are highly choreographed. The natureof these movements both in formation and flight path are complex andtake place at high speeds. Unfortunately, there are many situationswhere the presence of birds are highly problematic to human activitiesand infrastructure. For example, in the case of US Airways Flight 1549,colloquially referred to as the “Miracle on the Hudson,” the airplane onascent after takeoff struck a flock of geese and was forced to ditchinto the Hudson river. While keeping the skies clear over a small areahas been traditionally handled by trained birds of prey, for example thefamous Harris's Hawks of Wimbledon, this solution is not easilyscaleable nor feasible in many situations.

Drones, however, as opposed to biological solutions, aremass-producible, highly controllable, and can be stored long-term inmany different conditions with considerably less maintenance. Whilethere have been some attempts to utilize drones to scare away birds fromairports (for example RoBird of the Netherlands), these systems areoften remote controlled and designed to scare birds away. Both of thesefactors can cause significant issues, as both the ineffectiveness ofhuman piloting and unrefined scare tactics can lead to flocks of birdsto split up, which simply causes more problems in the form of multiplesub-flocks which must now be controlled. In turn, this requiresadditional UAVs or other interdiction methods (e.g. air cannons, lasers,biological solutions, etc.) which escalate cost and increase thelikelihood of strikes.

In contrast to the conventional “scare-tactics” which can split flocks,systems and methods described herein can mitigate these issues byutilizing autonomous drones to manipulate the movements of flocks ofbirds to guide them outside of a predefined geographic region withoutdisrupting the flock unit. This process is referred to as “herding,”because the flock is left intact. In many embodiments, multiple dronesacting in accordance with methods described herein can work in tandem toherd a single flock, or to herd multiple flocks at once. Autonomousdrones with appropriate sensing platforms can be pre-loaded with controlsoftware and deployed to safely remove birds in any number of scenariosnot limited to airports. However, for ease of description, airports willbe discussed in greater detail, and one of ordinary skill in the artwould understand that the systems and methods described herein couldeasily be deployed in alternative scenarios and locations.

In order to maintain flock integrity, in many embodiments, herdingprocesses include real-time modeling of a flock and the expectedresponse of the flock to external pursuers. In many embodiments, anumber of waypoints are calculated for the drone to maneuver to inrelation to the flock. The model of the flock dynamics can be used togenerate waypoints that avoid flock fragmentation and/or reduce theamount of physical space that the flock occupies (e.g. promote tighterflying formations). In numerous embodiments, multiple drones can beutilized, each with their own set of waypoints collectively calculatedto herd the flock. Flock herding systems are discussed in more detailbelow.

Flock Herding Systems

Flock herding systems are systems which prevent flocks of birds fromentering a pre-defined geographical region, or a “protected zone.” Inmany embodiments, a protected zone includes the airspace above aparticular region. As mentioned above, protected zones can be any numberof different regions, and are not defined by any particular geographicfeature. Drones can be utilized either alone or in tandem to keep birdsout of the protected zone. In many embodiments, drones have sensors toenable them to at least detect birds and/or other drones. However, innumerous embodiments, external sensors and/or sensor arrays are utilizedto track bird movements.

Turning now to FIG. 1 , a flock herding system in accordance with anembodiment of the invention is illustrated. Flock herding system 100includes a drone 110 and a flock herding control system 120. In manyembodiments, drones are quadcopters, however, any number of differentdrone platforms can be utilized, including, but not limited to, thosethat mimic birds of prey, fixed wing unmanned craft, single or othermulti-rotor craft, and or any other UAV as appropriate to therequirements of specific applications of embodiments of the invention.Flock herding control systems are computing platforms capable ofingesting bird position data and producing waypoints for drones tonavigate to in order to remove flocks from a protected zone. In manyembodiments, flock herding control systems are implemented on localcomputing platforms such as a server or personal computer. However,flock herding control systems can be implemented using cloud computingarchitecture and/or any number of different computing platforms asappropriate to the requirements of specific applications of embodimentsof the invention.

Flock herding system 100 further includes a bird location sensor 130. Inmany embodiments, bird location sensors 130 are avian radars. However,bird location sensors can be any sensor platform capable of reportingthe location of birds in a protected zone in at least near real-time. Innumerous embodiments, flock herding systems include additional sensorsfor tracking other environmental objects or conditions (collectively“environmental sensors”). For example, if an airport is in a protectedzone, additional environmental data can be produced regarding theposition of nearby aircraft. Further, environmental conditions such aswind speed, humidity, temperature, cloud cover, time, date, and anyother environmental information can be captured by sensors and providedto flock herding control systems for use in waypoint generation and/orother control functions.

Flock herding system 100 additionally includes an interface device 140.Interface devices are any device capable of enabling a user tocommunicate with a flock herding control system or a drone. In manyembodiments, interface devices are personal computers, but can be anynumber of digital communication devices such as, but not limited to,smart phones, smart TVs, tablet computers, and/or any other digitalcommunication device as appropriate to the requirements of specificapplications of embodiments of the invention. Components of the flockherding system 100 communicate via a network 150. In many embodiments,the network is a composite network made of multiple differentinterconnected networks which can be implemented using the same ordifferent communications protocols. For example, networks can be wired,wireless, or a combination thereof. For example, in some embodiments,flock herding control systems communicate with bird location sensors ona particular network, and the flock herding control system provides datato the drones via a separate network. One of ordinary skill in the artwould appreciate that any different number of system architecturesand/or communications architectures can be utilized as appropriate tothe requirements of specific applications of embodiments of theinvention, including, but not limited to, those that utilize differentenvironmental sensors and/or different numbers of drones.

Turning now to FIG. 2 , a conceptual block diagram of a flock herdingcontrol system in accordance with an embodiment of the invention isillustrated. Flock herding control system 200 includes a processor 210.Processors can be any number of logic processing circuitries including,but not limited to central processing units (CPUs), graphics processingunits (GPUs), field programmable gate arrays (FPGAs), applicationspecific integrated circuits (ASICs), and/or any other logic processingcircuitry capable of computation. In many embodiments, more than oneprocessor is utilized. Flock herding control system 200 further includesan input/output interface 220. Input/output interfaces enablecommunication to other parts of flock herding systems.

Flock herding control system 200 further includes a memory 230. Memoriescan be implemented using volatile memory, non-volatile memory, or acombination of both. Memory 230 contains a flock herding application232. Flock herding applications instruct the processor to generatewaypoints for drones to navigate to in order to herd flocks away fromprotected zones. In many embodiments, memory 230 further contains birdposition data 234 and/or environment data 236. As noted above, birdposition data describes the 3D coordinate location of birds in range ofthe protected zone, and environment data describes various environmentalconditions.

In many embodiments, flock herding control systems are implemented onstandard computing platforms. However, flock herding control systems canbe integrated into drones so long as the drone platform has sufficientcomputing capabilities. As one can appreciate, any number of differentimplementations of system architectures and flock herding controlsystems can be utilized to generate waypoints that enable drones tosuccessfully herd flocks away from a protected zone as appropriate tothe requirements of specific applications of embodiments of theinvention. Processes for generating waypoints are discussed below.

Generating Waypoints for Flock Herding

As noted above, a key problem with conventional bird dispersaltechniques is the fracturing of flocks into individual birds and/orsubflocks. This creates significant inefficiencies as the fracturedunits are dispersed over a wider area and then must each individually bedealt with. In contrast, processes described herein can generatewaypoints that, which when navigated to by a drone, will herd a flockaway from a protected zone without fracturing the flock. This leads canlead to an increase in efficiency at least by requiring fewer drones,quicker dispersal times, and more efficient use of resources generally.

By way of example, a high-level conceptual illustration of the effect ofa flock herding process in accordance with an embodiment is illustratedin FIG. 3 . In this case an airport is within a cylindrical protectedzone with radius R_(PZ). However, protected zones are not required to becylindrical, and be any number of shapes, for example, a sphere, a cube,a polyhedron, or a composite shape. When an flock on an intruding pathis detected, a drone is dispatched to herd the flock away from theprotected zone.

Turning now to FIG. 4 , a high-level flowchart for a flock herdingprocess in accordance with an embodiment of the invention isillustrated. Process 400 includes detecting (410) an intrusion. In manyembodiments, the intrusion is determined when a flock enters a protectedzone. However, intrusion can be determined when a flock is determined tobe on a trajectory that will lead to the protected zone. In numerousembodiments, this is accomplished by having a zone outside the protectedzone where flocks are tracked to determine if herding is necessary.

To accomplish successful herding, the “m-waypoint herding algorithm” hasbeen developed. At a high level, a drone interacts with the intrudingflock by positioning itself sequentially at a periodically refreshed setof m points. The inherent stability of the flocking dynamics areleveraged to prevent fragmentation, and the m points can be selected tomaximize the deflection of the flock's flight path.

Turning now to FIG. 4 , a particular implementation of a flock herdingprocess utilizing the m-waypoint herding algorithm in accordance with anembodiment of the invention is illustrated. Process 400 further includesobtaining (420) flock position data. In numerous embodiments, the flockposition data is obtained from the bird location sensor. Flock positiondata can described the location of individual birds, or the location ofa flock as a whole. In many embodiments, individual bird locations canbe aggregated into a single unit representing a flock. For example, bymeasuring proximity, speed, direction, and/or any other positioningmeasurement, birds flying in a coherent pattern and/or in proximity canbe grouped. A waypoint is generated (430) and a drone is navigated (440)to the waypoint. If the flock is not yet out of the protected zone(450), then updated flock position data can be obtained and a newwaypoint can be generated for the drone to navigate to until the flockhas been successfully removed. In numerous embodiments, waypoints aregenerated as a sequential set instead of one at a time, and the dronenavigates to each point in order. In a variety of embodiments, the setof waypoints is continuously updated. In many embodiments, multiple setsof waypoints can be generated for multiple drones to work in tandem toherd the same flock, and/or different flocks.

In order to generate the waypoints, an understanding of flock dynamicscan be modeled from which the basis of the m-waypoint herding algorithmcan be formulated. Consider a flock of n (usually >>1) birds. Theposition and the velocity of the i^(th) bird are denoted by x_(i)∈

³ and v_(i) ∈

³, respectively. The vector between the i^(th) and j^(th) birds isdenoted by r_(ij)=x_(j)−x_(i). The subscript ‘p’ is used for thepursuer, so that x_(p) denotes its position vector, andr_(pi)=x_(i)−x_(p) is the vector from the pursuer to the i^(th) bird.Given a vector r, the unit vector along r is denoted by {circumflex over(r)}.

Let R_(com) denote the communication range for interaction between twobirds. The neighborhood set of the i^(th) bird is defined using thestandard Euclidean distance as,

_(i) ={j≤n:|∥x _(i) −x _(j) ∥≤R _(com)}.  (1)and it is assumed that two birds interact only if their distance is lessthan or equal to R_(com). The set of birds (ν) thus forms a graph G=(ν,ε) whereν={1,2, . . . ,n},ε={(i,j)∈ν×ν|j∈

⊆

_(i)}, card(ε)=p  (2)

Note that if

=

_(i) for all i, the graph is undirected. Otherwise, it is directed. Theincidence matrix of the graph is denoted as B_(c)∈

^(n×p). Recall that the (i,j)^(th)b entry of B_(c) is 1 if the j^(th)edge is incoming at i, −1 if outgoing at i, and 0 otherwise. Anundirected edge is treated as two separate directed edges in thisformulation of B_(c). Δ(ν) denotes the diagonal matrix formed by theelements of the tuple ν. Finally, the centroid and the radius of theflock are defined as

$\begin{matrix}{{x_{cg} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}x_{i}}}},{R_{F}\overset{\Delta}{=}{\max_{i}{{x_{i} - x_{cg}}}}}} & (3)\end{matrix}$A. Flock Dynamics

The dynamics of an individual (i^(th)) bird are described by anonlinear, second-order differential equation based on Reynolds' rulesand augmented with an evasive response:

$\begin{matrix}{\mspace{79mu}{{\overset{.}{x}}_{i} = v_{i}}} & (4) \\{{\overset{.}{v}}_{i} = {{k_{s}{\sum\limits_{j \in \mathcal{J}_{i}}^{\;}{\left( {1 - \frac{R_{safe}^{3}}{{r_{ij}}^{3}}} \right)r_{ij}}}} + {k_{a}{\sum\limits_{j \in \mathcal{J}_{i}}^{\;}\left( {v_{j} - v_{i}} \right)}} + {k_{g}\left( {v_{d} - v_{i}} \right)} + {k_{p}{H\left( r_{pi} \right)}}}} & \;\end{matrix}$where R_(safe) denotes the steady-state distance between two adjacentmembers of the flock; v_(d) denotes the pre-selected, steady velocity ofeach member of the flock; and H(r_(pi))=H(x_(i)−x_(p)) denotes theevasive response to the pursuer. It is assumed that v_(d) lies in thehorizontal plane.

FIG. 5 summarizes important distance variables that appear in the modelin accordance with an embodiment of the invention. The terms R_(safe)and R_(com) have been defined already. The terms R_(fear) and R_(agg)(<R_(fear)) refer to critical distances in the bird-pursuer interaction,and it is assumed that both of these quantities are known for a givenflock. The flock responds to a pursuer only when the pursuer is within aradius R_(fear) of a member of the flock. Within R_(fear), the evasivebehaviour generally takes two different forms:

-   -   If the pursuer is within [R_(agg), R_(fear)], the bird tries to        accelerate radially away from the pursuer.    -   If the pursuer is within R_(agg), the bird tries to out-maneuver        the pursuer by turning and/or climbing rapidly.

Since the out-turning or 3-D behavior can have adverse implications forthe stability of the flock, R_(agg) can be treated as a lower bound forthe permissible distance between the pursuer and the flock. H(⋅) can bemodeled as follows:

$\begin{matrix}{{H\left( r_{pi} \right)} = \left\{ \begin{matrix}{\frac{r_{pi}}{{r_{ij}}^{3}},} & {{{if}\mspace{14mu}{r_{pi}}} < {R_{fear}\mspace{14mu}{and}\mspace{14mu} r_{pi}^{T}{\overset{.}{r}}_{pi}} < 0} \\0 & {otherwise}\end{matrix} \right.} & (5)\end{matrix}$and a restriction that ∥r_(pi)∥ R_(agg)+ϵ for all i, where ϵ>0 is anarbitrarily-chosen constant can be imposed.B. Herding Objectives

As noted above, the protected zone (PZ) is the spatial volume that needsto be kept free of birds. For illustrative purposes, in this example, PZis assumed to be cylindrical with radius R_(PZ). However, in manyembodiments, the PZ can be other shapes. For modeling purposes, it canbe assumed that birds fly along a constant, pre-selected heading,{circumflex over (v)}_(d), when not disturbed by a pursuer. This is infact true, for instance, for flocks flying to migration groundsconsiderably far from PZ. It is clear that the pursuer only needs tomove the flock to a point from which its flight path (along {circumflexover (v)}_(d)) would no longer cross PZ. This target point is denoted asx_(div)(t), noting that its exact position may change with time asdescribed presently.

The global coordinate frame is defined with its origin placed at thecentre of PZ, and its axes pointing north, east and up, respectively.Let h(t) denote the instantaneous altitude of the flock. First, twovertical lines (in the global frame) can be constructed which serve asloci for all possible target points. Recall that v_(d) in (4) is thepredetermined nominal flock velocity which satisfies {circumflex over(k)}^(T)v_(d)=0 where {circumflex over (k)} is the unit vector pointingupwards in the global frame. Unit vector {circumflex over (v)}_(d) ^(⊥)is defined which satisfies v_(dt){circumflex over (v)}_(d)^(⊥)={circumflex over (k)}^(T){circumflex over (v)}_(d) ^(⊥)=0. The twocandidate loci are then given by l _(cand,±)(a)=±(R_(PZ)+s){circumflexover (v)}_(d) ^(⊥)+a{circumflex over (k)} where a∈

, and s>R_(F) is a suitably chosen constant. R_(div)

R_(PZ)+s is denoted. Of these candidate loci, the one closest to x_(cg),at t=0 (the moment at which the herding commences) is selected, asillustrated in FIG. 6 in accordance with an embodiment of the invention.Finally, at each t, set the instantaneous target waypoint on the chosenlocus as x_(div)=(+/−)R_(div){circumflex over (v)}_(d)^(⊥)+h(t){circumflex over (k)}. If the two loci are equidistant fromx_(cg), then any of the two loci can be chosen: however, this choice maypersist for the remainder of the herding operation. This selection ofherding target points is a logical extension from 2D to 3D when theherding algorithm does not seek to control the altitude of the flock.The objective of the herding algorithm is to prescribe a trajectoryx_(p)(t) for the pursuer which enables it to: (1) move the CG of theflock to x_(d), in finite time; (2) avoid fragmentation of the flock;and (3) keep the flock outside the protected zone PZ during the courseof herding.

C. Stability Analysis

In order to avoid fracturing the flock, conditions for flock stabilitycan be modeled. These conditions can be used to select the gains and theunderlying graph in (4) for analytical investigation when sufficientempirical data is not available to determine these parameters. Theresulting robustness of the flock implies that the pursuer is free toapproach the flock from any direction. As long as it maintains a certainminimum distance between itself and the flock, the flock will notundergo fragmentation.

C.I. Snapshot Dynamics, Steady States, and Linearization

To model snapshot dynamics, steady states, and linearization, two columnvectors x, v∈

^(3n), formed by permuting the components of x_(i) and v_(i),respectively, ∀i:x=[x ₁₁ , . . . ,x _(n1) ,x ₁₂ , . . . ,x _(n2) ,x ₁₃ , . . . ,x_(n3)]^(T)are defined, and v is constructed in a similar manner. There exists apermutation matrix, denoted by P_(3n)∈

^(3n×3n), such that

${x = {P_{3\; n}\begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{n}\end{bmatrix}}},{\begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{n}\end{bmatrix} = {P_{3\; n}^{- 1}x}}$

Ignoring the pursuer-dependent terms, the flocking dynamics (4) can bewritten as{dot over (x)}=v{dot over (v)}=−k _(a)(I ₃ ⊗D _(c) B _(c) ^(T))v+k _(g)(v _(d)⊗1_(n)−v)−k _(s)(I ₃ ⊗D _(c) Q(x)B _(c) ^(T))x  (6)where D_(c)(i,j)=−1 if the j^(th) edge is incoming at i, andD_(c)(i,j)=0 otherwise. Here, I₃ the 3×3 identity matrix, while ⊗denotes the Kronecker product. D_(c) can be replaced with B_(c)/2 if thegraph is undirected. The matrix Q(x)∈

^(p×p) is a diagonal matrix satisfying Q(j,j)=1−R_(safe) ³/∥e_(j)∥₃.

In order to analyze the stability of the flock, a change of coordinatescan be made: {tilde over (x)}_(i)(t)=x_(i)(t)−v_(d)t, {tilde over(v)}_(i)(t)=v_(i)(t)−v_(d). Since Q(x)=Q({tilde over (x)}), flockingdynamics can be written as{tilde over ({dot over (x)})}={tilde over (v)}{tilde over ({dot over (v)})}=−k _(s)(I ₃ ⊗D _(c) Q({tilde over (x)})B_(c) ^(T)){tilde over (x)}−k _(a)(I ₃ ÐD _(c) B _(c) ^(T)){tilde over(v)}−k _(g) {tilde over (v)}  (7)

This is a nonlinear, autonomous system whose equilibrium solutions aregiven by [{tilde over (x)}⁰, 1_(n)⊗0₃], where {tilde over (x)}⁰satisfies (I₃⊗D_(c)Q({tilde over (x)}⁰)B_(c) ^(T)){tilde over (x)}⁰=0.

Given an equilibrium configuration {tilde over (x)}⁰, the family ofequilibrium solutions obtained by rigid-body translation and rotationthe flock are defined byS({tilde over (x)} ⁰)={{tilde over (x)}|{tilde over (x)}=b⊗1_(n) +αP_(3n)(I _(n) ⊗J)P _(3n) ⁻¹ {tilde over (x)} ⁰}  (8)where b∈

³, J∈ so (3), and α∈

.

Let δ· denotes a small perturbation in the corresponding variable, ande_(i) ⁰ is the steady-state i^(th) edge vector. Then, the linearizationof (7) about an equilibrium point is given by

${{\delta\overset{.}{x}} = {\delta\upsilon}}{{\delta\overset{.}{\upsilon}} = {{{- \frac{3}{R^{2}}}\left( {{I_{3} \otimes D_{c}}{\Delta\left( {{e_{1}^{0^{\top}}\delta e_{1}},\ldots,{e_{p}^{0^{\top}}\delta e_{p}}} \right)}B_{c}^{\top}} \right){\overset{\sim}{x}}^{0}} - {\left( {{k_{a}\left( {{I_{3} \otimes D_{c}}B_{c}^{\top}} \right)} + {k_{g}I_{3n}}} \right){\delta\upsilon}}}}$

These equations can be written compactly as

${\begin{bmatrix}{\delta\overset{.}{x}} \\{\delta\overset{.}{\upsilon}}\end{bmatrix} = {G\begin{bmatrix}{\delta x} \\{\delta\upsilon}\end{bmatrix}}},$where the Jacobian G is given by

$\begin{matrix}{{G = \begin{bmatrix}0_{3n} & I_{3n} \\{{- k_{s}}A} & {- B}\end{bmatrix}},} & (9)\end{matrix}$and

$\begin{matrix}{{B = {{k_{a}\left( {{I_{3} \otimes D_{c}}B_{c}^{\top}} \right)} + {k_{g}I_{3n}}}}{{A\delta x} = {\frac{3}{R^{2}}\left( {{I_{3} \otimes D_{c}}{\Delta\left( {{e_{1}^{0^{\top}}\delta e_{1}},\ldots,{e_{p}^{0^{\top}}\delta e_{p}}} \right)}B_{c}^{\top}} \right){\overset{\sim}{x}}^{0}}}} & (10)\end{matrix}$C.II. Spectral Properties of A and G

In order to determine the spectral properties of A and G:

Lemma 1. The null space of A defined in (10) is a superset ofeigenvectors which correspond to a rigid body translation and rotationof the flock; i.e.,

(A)⊇

≙span([1,0,0]^(T)⊗1_(n)[0,1,0]^(T)⊗1_(n),[0,0,1]^(T)⊗1_(n) ,P _(3n)(I _(n) ⊗J ₁)P _(3n) ⁻¹ {tilde over (x)} ⁰ ,P_(3n)(I _(n) ⊗J ₂)P _(3n) ⁻¹ {tilde over (x)} ⁰,P _(3n)(I _(n) ⊗J ₃)P _(3n) ⁻¹ {tilde over (x)} ⁰)where J₁, J₂, J₃ form a basis for so (3).Proof: It is evident from (10) that

(A) is a superset of the vectors δx which satisfy (I_(n)⊗B_(c) ^(T))δx=0or e⁰ _(i) ^(T)δe_(i)=0 ∀i={1, . . . , p}. The former condition impliesthat δx=k₁[1, 0, 0]^(T)⊗1_(n)+k₂[0, 1, 0]^(T)⊗1_(n)+k₃[0, 0,1]^(T)⊗1_(n), where k₁, k₂, and k₃ are arbitrary constants. The lattercondition is satisfied by δx_(i)=Σ_(j=1) ³k′_(j)J_(j)x_(i), for allconstants k′_(j).Assumption 1. The matrix A has exactly six eigenvalues which are equalto 0. Therefore,

_(u)(A)=

(A).Assumption 1 is satisfied by connected undirected graphs and by stronglyconnected and balanced digraphs.Assumption 2. The Jacobian G has the following properties:

-   -   1. All eigenvalues of G have non-positive real parts.    -   2. The algebraic and geometric multiplicities of the zero        eigenvalue are equal to each other.    -   3. The null space        (G)=[n^(T), 0_(1×3n)]^(T), where n∈        (A)=        _(u)(A).        Assumption 2 always holds for an undirected graph satisfying        Assumption 1.        C.III. Stability Analysis

To perform the stability analysis, the center manifold theorem can beused to show that trajectories which start away from S({tilde over(x)}⁰), at least locally, converge to S({tilde over (x)}⁰) at anexponential rate. The proof proceeds in two steps. First, it is shownthat there exists a center manifold when Assumption 2 is satisfied. Itis noted that the set S({tilde over (x)}⁰) is also a center manifold. Inthe second step, the property that center manifolds are arbitrarilyclose to each other can be used to complete the proof.

Lemma 2. Suppose that the flocking dynamics satisfy Assumption 2. Then,for every steady state {tilde over (x)}⁰, there exists an invariantmanifold ε^(c) ≙ε^(c)({tilde over (x)}⁰) such that any trajectorystarting around ε^(c) converges at an exponential rate to ε^(c).

Proof: From Assumption 2, it is known that the Jacobian G, evaluated at{tilde over (x)}⁰, has six zero eigenvalues and the other eigenvaluesare negative. Thus, it follows that from the center manifold theoremthat there exists a six-dimensional center manifold ε^(c) such thattrajectories starting outside ε^(c) converge at an exponential rate toε^(c).Theorem 1. The set S({tilde over (x)}⁰) is exponentially stable.Proof: It is noted that S({tilde over (x)}⁰) is itself a center manifoldof the dynamics. From Lemma 2, it is known that trajectories starting ina neighborhood of x⁰ converge at an exponential rate to the centermanifold ε^(c)({tilde over (x)}⁰). Recall that any two center manifoldsabout an equilibrium point {tilde over (x)}⁰ differ by transcendentallysmall terms, and every equilibrium point in the vicinity of {tilde over(x)}⁰ lies on every center manifold ε^(c)({tilde over (x)}⁰). Thus, atrajectory starting in a neighborhood of {tilde over (x)}⁰ convergesexponentially fast to S({tilde over (x)}⁰).

The discussion above shows that S({tilde over (x)}⁰) is exponentiallystable, and therefore, robust to small perturbations. In the context ofherding, this implies that, for every direction in relation to the flock(or its center), there exists a nontrivial set of pursuer positions forwhich the underlying graph of the flock is preserved. Below areanalytical formulae for estimating a set of permissible pursuerpositions based purely on local topological considerations.

In addition to stability, for the purpose of herding, it is useful tohave the property of time-scale separation between the synchronizationof the flocking dynamics to S({tilde over (x)}⁰) on the one hand, andthe translational dynamics of the center of gravity of the flock (whosetime constant, as shown in above, is 1/k_(g)) on the other. This can beachieved by a suitable choice of k_(s) and k_(a) for a given graphtopology.

D. Estimation of the Permissible Set of Pursuer Positions

The objective of this section is to derive approximations for theapproach distance between a pursuer and the flock, based purely on thelocal interaction between the pursuer and the birds in a flock. Thisestimate supersedes the distance R_(agg) in FIG. 2 from the point ofview of preserving the local topology of the flock, and is used in belowto determine the feasible positions for the pursuer in relation to theflock.

The influence exerted by the pursuer can cause one of two effects inextreme circumstances. In one case, the pursuer can push a bird towithin an unacceptable distance of its neighbor. This scenario is likelyin dense flocks, as shown in FIG. 7 . In the second case, the pursuercan cause the link between two neighbors to be broken by pushing thembeyond their communication distance R_(com). This scenario, illustratedin FIG. 8 , is generally more plausible in sparse flocks. The minimumpermissible distance between two birds is written as c R_(safe), wherec<1 is a constant that depends on the species of birds and the pursuerrepresented by the drone.

D.I. Maintaining Minimum Allowable Distance Between Neighbors

Consider the situation wherein a bird on the boundary of the flock(labeled ‘1’) is engaged by the pursuer. The bird tries to move awayradially from the pursuer, and into the flock. In the worst case, thebird's evasive path points in the direction of a neighboring bird (‘2’)and there is force from other neighboring birds pulling ‘1’ away from‘2.’ This is depicted in accordance with an embodiment of the inventionin the dense flock of FIG. 7 .

A conservative estimate for the minimum approach distance between thepursuer and the bird ‘1’ can be found from the local neighborhood ofFIG. 7 . This is a repelling, “outward-pointing” (with respect to theflock) force on ‘1’ due to ‘2’ is countered by the inward-pointing forceon ‘1’ arising as a consequence of repulsion by the pursuer. Balancingthese two forces when the distance between ‘1’ and ‘2’ is c R_(safe)(thereby ensuring that the minimum approach distance is no less thanR_(safe)) gives the minimum approach distance ∥r_(p1)∥_(min):

$\begin{matrix}{{\frac{k_{p}}{{r_{p1}}_{\min}^{2}} = {k_{s}{❘{{cR}_{safe}\left( {1 - \frac{R_{safe}^{3}}{\left( {cR_{safe}} \right)^{3}}} \right)}❘}}}{\left. \Longrightarrow{r_{p1}}_{\min} \right. = \sqrt{\frac{c^{2}k_{p}}{\left( {1 - c^{3}} \right)k_{s}R_{safe}}}}} & (11)\end{matrix}$D.III. Preserving Communication Between Neighboring Birds

In sparse flocks, it is commonplace to find a linear topology, involvingtwo or three agents forming an angle or a straight line with no othercommon neighbors. Consider a single edge, as shown in FIG. 9 inaccordance with an embodiment of the invention, involving just twobirds, and assume that the pursuer approaches along the perpendicularbisector of the edge. If R_(com) is the maximum permissible distancebetween two boundary agents for the underlying graph to be preserved,the following condition for the minimum permissible ∥r_(p1)∥ is obtained

$\begin{matrix}{{{\frac{k_{p}}{{r_{p1}}^{2}} \times \frac{R_{com}}{2{r_{p1}}}} = {{k_{s}\left( {1 - \frac{R_{safe}^{3}}{R_{com}^{3}}} \right)}R_{com}}}{\left. \Longrightarrow{r_{p1}}_{\min} \right. = \left( \frac{k_{p}}{2{k_{s}\left( {1 - \frac{R_{safe}^{3}}{R_{com}^{3}}} \right)}} \right)^{1/3}}} & (12)\end{matrix}$Below, example estimates for ∥r_(p1)∥_(min) are used as part of themotion planning strategy for the pursuer.Definition 1. Define a set X_(p) of permissible pursuer positions inrelation to the flock as:

$\begin{matrix}{{\chi_{p} = \left\{ x_{p} \middle| {R_{\min} \leq {\min\limits_{i \in V}{{x_{p} - x_{i}}}} \leq R_{fear}} \right\}},} & (13)\end{matrix}$where R_(min) is the maximum of R_(agg) and either (11) or (12)depending on the topology of the flock.E. The M-Waypoint Herding AlgorithmThe above established conditions for stability and robustness of theflock, and derived approximations for the approach distance between theflock and the pursuer. In this section, the problem of herding flockswithout fracturing them is solved using waypoints generated by modelingthe ensemble behavior of the flock. A version of the m-waypoint herdingalgorithm in accordance with an embodiment of the invention isillustrated in FIG. 10 . A set of objectives that the pursuer'strajectory, x_(p)(t), must satisfy in order herd the flock successfullyare provided below. The m-waypoint algorithm is then obtained as anapproximate, but efficient, solution to the problem of ensuring thatx_(p)(t) meets these objectives.E.I. Formulation of the Herding Problem

The coordinates of the flock's centroid and its velocity are given by

${x_{cg} = {\frac{1}{n}{\sum_{i = 1}^{n}x_{i}}}},{\upsilon_{cg} = {\frac{1}{n}{\sum_{i = 1}^{n}{\upsilon_{i}.}}}}$Using (4) and (5), obtain

$\begin{matrix}{{{\overset{.}{x}}_{cg} = \upsilon_{cg}}{{\overset{.}{\upsilon}}_{cg} = {{k_{g}\left( {\upsilon_{d} - \upsilon_{cg}} \right)} + {F_{p}\left( x_{p} \right)} + {f\left( {x,\upsilon} \right)}}}{{F_{p}\left( x_{p} \right)}\overset{\Delta}{=}{\frac{k_{p}}{n}{\sum\limits_{i \in \mathcal{N}_{p}}{H\left( r_{pi} \right)}}}}{{\overset{.}{x}}_{p} = u_{p}}} & (14)\end{matrix}$where

denotes the subset of the flock that lies within R_(fear) of thepursuer; the pursuer's velocity, u_(p), is a control variable for theherding algorithm. The term f(x,v) in zero when the graph is undirected(since 1_(n) ^(T)B_(c)=0) or if it has synchronized to a steady stateconfiguration. This would occur naturally when the flock synchronizes ona faster time scale than the dynamics of its CG. Since the goal is forthe flock to point in the direction of the herding target point, itsuffices to ensure that its velocity normal to an axis pointing towardsx_(div) is driven to zero. Definer _(dc)(t)≙x _(div)(t)−x _(cg)(t),q≙r _(dc) ×v _(cg)  (15)

Recall that {circumflex over (k)}^(T)x_(div)(t)={circumflex over(k)}^(T)x_(cg)(t)=h(t) (the altitude of the flock). Since x_(div)(t)lies on a fixed vertical line for all t, then {circumflex over (k)}×{dotover (x)}_(div)(t)=0. Since {dot over (r)}_(dc)(t)={dot over(x)}_(div)(t)−v_(cg)(t), the following expression is obtained for thedynamics of q:{dot over (q)}=r _(dc) ×{dot over (v)} _(cg) +{dot over (x)} _(div) ×v_(cg)=−k _(g) q+k _(g)(r _(dc) ×v _(d))+r _(dc) ×F _(p)(x _(p))+r _(dc)×f(x,v)+{dot over (x)} _(div) ×v _(cg)Next, define q_(k) ≙{circumflex over (k)}^(T)q. Note that {circumflexover (k)}^(T)({dot over (x)}_(div)×v_(cg))=v_(cg) ^(T)({circumflex over(k)}×{dot over (x)}_(div))=0. Thus, the dynamics of q_(k) are given by{dot over (q)} _(k) +k _(g) q _(k) ={circumflex over (k)} ^(T)(r_(dc)×(k _(g) v _(d) +f(x,v))+r _(dc) ×F _(p)(x _(p)))  (16)

Denote the amount of deviation that can be produced by placing thepursuer at x_(p) using the right hand-side of (16):ρ(x _(p))≙{circumflex over (k)} ^(T)(r _(dc)×(k _(g) v _(d) +f(x,v))+r_(dc) ×F _(p)(x _(p)))  (17)If ρ(x_(p))=0 for all t, then it is possible to deduce from (16) thatq_(k)→0 and the CG of the flock moves in a straight line (represented asa solid line in FIG. 9 in accordance with an embodiment of theinvention) towards x_(div) (from the definition of q_(k)). Further, notethat the distance and the time required for herding could be reduced, ascompared to moving the flock along the solid line, by pushing it rapidlytowards the dashed line passing through x_(div) in FIG. 9 . The dashedline is parallel to v_(d), and once the flock's CG reaches it, theherding algorithm can safely terminate. This requires that q_(k)≥0uniformly (or q_(k)≤0 uniformly, depending on the choice of x_(div))during the course of herding. Without loss of generality, it can beassumed that x_(div) is chosen such that sign({circumflex over(k)}^(T)(v_(d)×r_(dc)))>0 while commencing herding and set the controlobjective to maximizing ρ(x_(p))≥0 while sign({circumflex over(k)}^(T)(v_(d)×r_(dc)))>0 (see FIG. 9 ).E.II. Calculating m Waypoints

A solution to the aforementioned control problem is to compute thetrajectory of x_(p)(t) which maximizes ρ(x_(p)), and get the pursuer totrack it. It is known that a flock tends to deform into a concave shapelocally under persistent pressure from a pursuer, which is known to dentthe effectiveness of the pursuit. Furthermore, over-stressing one ormore birds continuously over an extended period of time carries the riskof the distressed birds attempting aggressive evasive maneuvers andfragmenting the flock. To avoid these problems, a sub-optimal approachcan be employed wherein the flock is engaged through different waypointsin a given timefranme.

The waypoints are chosen by sampling the set X_(p) from Definition 1.Sampling is preferred to statically defined waypoints because it allowsthe algorithm to be agnostic to the exact geometry of the flock. This isuseful when the flock is not necessarily best represented as a convexshape; for instance, star-shaped flocks and flocks with a curvilineargeometry. The set of m waypoints can be formally identified as follows.

Definition 2. The set X_(p) ^(m) is defined by construction. Thewaypoint selection algorithm, illustrated in accordance with anembodiment of the invention in FIG. 11 , samples X_(p) uniformly. Next,up to m waypoints with the highest deviation are identified such that notwo waypoints are within a prescribed distance, denoted δ_(w), of eachother. These waypoints constitute the set X_(p) ^(m).Even for a convex shaped flock, the set X_(p) ^(m) need not beconnected. This enables the use of random sampling for construction.F. Motion Planning for the Pursuer

The motion planner solves for the pursuer's velocity u_(p)(t) bycommanding one of two motions, as follows.

-   -   1. FLY: the pursuer takes the fastest path to the commanded        node. Collision avoidance is achieved using artificial potential        fields, an alternative to which is a real-time, online motion        planner.    -   2. ENGAGE: using a virtual leader-based approach, the pursuer        commands u_(p)(t) to maintain its position at the chosen        waypoint in relation to the flock's centroid for a        pre-determined duration τ_(e). The engagement is terminated if        the distance between two neighboring birds in        _(p) approaches the communication radius, R_(com).        G. Tuning the Herding Algorithm

In order to promote successful herding, a condition is derived in theform of the minimum allowable distance from the protected zone at whichthe herding must begin. If the herding commences beyond this distance,it will succeed with an elevated likelihood. Assume that f(x, v)=0;i.e., the underlying graph is undirected or the flock has synchronizedto a steady state.

To begin, ignore the dynamics of the pursuer, and recall that thecontrol objective is to maximize ρ(x_(p))≥0. Consider the conservativecase ρ(x_(p))=0 for all t during the course of herding. This case islimiting in that the distance at which the herding needs to be commencedcan be lesser when larger values of ρ(x_(p)) are attainable.

In order to achieve ρ(x_(p))=0, set x_(p)=x_(p)* whereF_(p)(x_(p)*)=−k_(g)({circumflex over (k)}^(T)({circumflex over(r)}_(dc)×v_(d)))({circumflex over (k)}×{circumflex over (r)}_(dc)).Since {circumflex over (r)}_(dc)=r_(dc)/∥r_(dc)∥, and v_(d) lie in thehorizontal plane, it follows that ∥F_(p)∥=k_(g)∥{circumflex over(r)}_(dc)×v_(d)∥. Let ρ_(F) denote the upper bound on ∥F_(p)∥, which isknown. It is clear then that the flock should be made to satisfy∥{circumflex over (r)}_(dc)×v_(d)∥<ρ_(F)/k_(g) ∀t, which, in turn, meansthat the flock must be kept outside the cone shown in FIG. 12 . This isan important insight: it shows that the herding is successful only if itbegins when the horizontal distance between the centre of the flock andthe center of PZ is greater than or equal to

$\begin{matrix}{{D_{p,\min}\overset{\Delta}{=}\frac{R_{div}}{\tan\theta_{\max}}},{\theta_{\max} = {\sin^{- 1}\left( \frac{\rho_{F}}{k_{g}{\upsilon_{d}}} \right)}},} & (18)\end{matrix}$where R_(div) was defined above.

While the solution x_(p)* appears to solve the problem in principle,there is the possibility that the trajectory x_(p)* may not be feasible.Furthermore, the motion planning algorithm adopted here requires thatthe pursuer engage with an entire “front” of the flock rather thanspecific individual birds. In particular, it means that the engagementbetween the pursuer and the flock takes place in pulses. Processesdescribed herein can be refactored in light of this pulsed interaction.

Let τ_(e) denote the duration of any given engagement between thepursuer and the flock, and let τ_(f) denote the time taken by the birdto fly between two waypoints. During the time τ_(f), the flock receivesno external stimulus and its velocity tends to align to v_(d). Assumingthat an engagement begins at time t₀, the dynamics of the flock in thetime interval [t₀, t₀+τ_(e)+τ_(f)] can be described via the switchingdynamics

$\begin{matrix}{{{\overset{.}{\upsilon}}_{cg} + {k_{g}\upsilon_{cg}}} = \left\{ \begin{matrix}{{k_{g}\upsilon_{d}^{❘❘}},} & {t \in \left\lbrack {t_{0},{t_{0} + \tau_{e}}} \right)} \\{{k_{g}\upsilon_{d}},} & {t \in \left\lbrack {{t_{0} + \tau_{e}},{t_{0} + \tau_{e} + \tau_{f}}} \right)}\end{matrix} \right.} & (19)\end{matrix}$where v_(d) ^(∥)=v_(d)+F_(p)/k_(g) (see FIG. 13 as an example inaccordance with an embodiment of the invention). When x_(p)=x_(p)*,v_(d) ^(∥) is along r_(dc) at every instant in time, which is why thesuperscript ∥ is used here. Assume that the flock switchesinstantaneously between two states. During the engagement phase, theflock moves at an angle θ to the original direction coinciding withv_(d). When the pursuer does not engage the flock, it moves along v_(d).This is depicted in FIG. 14 in accordance with an embodiment of theinvention.

The terms ∥v_(d) ^(∥)∥ and θ can be be estimated empirically since theprecise interaction between the flock and the pursuer is highlycase-specific. The magnitude of v_(d) ^(∥) depends on whether the flockis driven from the front or from the rear. Since the intent is for thenet motion to be perpendicular v_(d), it is reasonable to assume thatthe average force also acts perpendicular to v_(d). Thus,

${{v_{d}^{❘❘}} = \sqrt{{\upsilon_{d}}^{2} + \frac{{F_{p}}^{2}}{k_{g}^{2}}}},{\theta = {\cos^{- 1}\left( {{\upsilon_{d}}/{\upsilon_{d}^{❘❘}}} \right)}}$

Consequently,

Theorem 2. Given the approximate values of ∥v_(d) ^(∥)∥ and θ, therefined value of D_(p,min) is given by

$D_{p,\min} = {R_{div}\cot{\theta\left( {1 + \frac{\tau_{f}}{\tau_{e}}} \right)}}$Proof Let n_(s) denote the number of engagement pulses. From FIG. 14 ,it is apparent that there can be at most n_(s) pulses wherein the flockis not engaged with the pursuer. During each engagement pulse, referringto the geometric convention of FIG. 14 , the flock shifts through adistance ∥v_(d) ^(∥)∥ τ_(e) sin θ. Thus,

$n_{s} = \frac{R_{div}}{{\upsilon_{d}^{❘❘}}\tau_{e}\sin\theta}$Furthermore, during each engagement phase, the flock translateshorizontally through a distance ∥v_(d) ^(∥)∥ τ_(e) cos θ. During thenon-engagement phase, the horizontally translation is at moat∥v_(d)∥τ_(f). Thus, the minimum horizontal translation is given by

$\begin{matrix}{D_{p,\min} = {n_{s}\left( {{{\upsilon_{d}}\tau_{f}} + {{\upsilon_{d}^{❘❘}}\tau_{e}\cos\theta}} \right)}} \\{= {R_{div}\left( {{\frac{\upsilon_{d}}{\upsilon_{d}^{❘❘}}\frac{\tau_{f}}{\tau_{e}}\frac{1}{\sin\theta}} + {\cot\theta}} \right)}}\end{matrix}$Setting cos θ=∥v_(d)∥/∥v_(d) ^(∥)∥ yields

$D_{p,\min} = {R_{div}\cot{{\theta\left( {1 +_{\tau_{e}}^{\underset{¯}{\tau_{f}}}} \right)}.}}$Notice that (18) is recovered if when τ_(f)=0 and θ=θ_(max). Thiscompletes the proof. Using the above, flock flight dynamics can bemodeled and effective waypoints can be generated.

Although the present invention has been described in certain specificaspects, many additional modifications and variations would be apparentto those skilled in the art. It is therefore to be understood that thepresent invention can be practiced otherwise than specifically describedwithout departing from the scope and spirit of the present invention.Thus, embodiments of the present invention should be considered in allrespects as illustrative and not restrictive. Accordingly, the scope ofthe invention should be determined not by the embodiments illustrated,but by the appended claims and their equivalents.

What is claimed is:
 1. An autonomous bird flock herding system,comprising: at least one bird location sensor; an unmanned aerialvehicle; and a flock herding control system, comprising: a processor;and a memory, the memory containing a flock herding application, wherethe flock herding application directs the processor to: obtain birdposition data from the at least one bird location sensor, where the birdposition data describes the location of birds in a flock of birds, wherethe flock of birds forms a non-convex hull; determine if the flock ofbirds will enter a protected zone; generate a set of waypoints forpositioning the unmanned aerial vehicle using a predictive flockdynamics model, where for each waypoint in the set of waypoints, theunmanned aerial vehicle is predicted to not split the flock when at thewaypoint; uniformly sample the set of waypoints to produce a subsampleof waypoints; score each given waypoint in the subsample of waypointsbased on whether or not placing the unmanned aerial vehicle at the givenwaypoint will move the flock of birds in a desired direction and thedistance to other waypoints in the subsample of waypoints; sort thesubsample of waypoints based on score; select a waypoint from the sortedsubsample of waypoints having the highest score; instruct the unmannedaerial vehicle to navigate to the selected waypoint such that the flockof birds will, in response to the presence of the unmanned aerialvehicle at the at least one waypoint, change trajectory away from theprotected zone.
 2. The autonomous bird flock herding system of claim 1,wherein the flock of birds will, in response to the presence of theunmanned aerial vehicle at the selected waypoint maintain integrity suchthat a centroid of the flock and a shape of the flock are maintained andcommunication links between any two neighboring birds of the flock arenot broken.
 3. The autonomous bird flock herding system of claim 1,wherein the protected zone is a cylinder.
 4. The autonomous bird flockherding system of claim 1, wherein the protected zone is over anairport.
 5. The autonomous bird flock herding system of claim 1, whereinthe at least one bird location sensor is an avian radar.
 6. Theautonomous bird flock herding system of claim 1, further comprising atleast one environmental sensor; and the flock herding applicationfurther directs the processor to: obtain environment data describingenvironmental conditions proximal to the protected zone from the atleast one environmental sensor; and generate the set of waypoints usingthe environment data.
 7. The autonomous bird flock herding system ofclaim 6, wherein the environment data describes the position ofairplanes.
 8. The autonomous bird flock herding system of claim 6,wherein the environment data describes wind speed.
 9. The autonomousbird flock herding system of claim 1, wherein the flock herdingapplication further directs the processor to: obtain updated birdposition data; update the set of waypoints based on the updated birdposition data; uniformly sample the updated set of waypoints to producean updated subsample of waypoints; score each specific waypoint in theupdated subsample of waypoints based on whether or not placing theunmanned aerial vehicle at the specific waypoint will move the flock ofbirds in a desired direction and the distance to other waypoints in theupdated subsample of waypoints; sort the updated subsample of waypointsbased on score; and select a new waypoint from the sorted, updatedsubsample of waypoints having the highest score.
 10. The autonomous birdflock herding system of claim 1, further comprising: a second unmannedaerial vehicle; and wherein the flock herding application furtherdirects the processor to: generate a second set of waypoints using thepredictive flock dynamics model; uniformly sample the second set ofwaypoints to produce a second subsample of waypoints; score eachspecific waypoint in the second subsample of waypoints based on whetheror not placing the unmanned aerial vehicle at the specific waypoint willmove the flock of birds in a desired direction and the distance to otherwaypoints in the second subsample of waypoints; sort the secondsubsample of waypoints based on score; select a second waypoint from thesorted second subsample of waypoints having the highest score; andinstruct the second unmanned aerial vehicle to navigate to the selectedsecond waypoint such that the flock of birds will, in response to thepresence of the second unmanned aerial vehicle at the given at least onewaypoint, change trajectory away from the protected zone.
 11. A methodfor the autonomous herding of flocks of birds comprising: obtaining,using a flock herding control system, bird position data from at leastone bird location sensor, where the bird position data describes thelocation of birds in a flock of birds, where the flock of birds forms anon-convex hull; determining, using the flock herding control system, ifthe flock of birds will enter a protected zone; generating, using theflock herding control system, a set of waypoints using a predictiveflock dynamics model for an unmanned aerial vehicle, where for eachwaypoint in the set of waypoints, the unmanned aerial vehicle ispredicted to not scars split the flock when at the waypoint; uniformlysampling the set of waypoints to produce a subsample of waypoints;scoring each given waypoint in the subsample of waypoints based onwhether or not placing the unmanned aerial vehicle at the given waypointwill move the flock of birds in a desired direction and the distance toother waypoints in the subsample of waypoints; sorting the subsample ofwaypoints based on score; selecting a waypoint from the sorted subsampleof waypoints having the highest score; and instructing, using the flockherding control system, the unmanned aerial vehicle to navigate to theselected waypoint such that the flock of birds will, in response to thepresence of the unmanned aerial vehicle at the at least one waypoint,change trajectory away from the protected zone.
 12. The method for theautonomous herding of flocks of birds of claim 11, wherein the flock ofbirds will, in response to the presence of the unmanned aerial vehicleat the selected waypoint maintain integrity such that a centroid of theflock and a shape of the flock are maintained and communication linksbetween any two neighboring birds of the flock are not broken.
 13. Themethod for the autonomous herding of flocks of birds of claim 11,wherein the protected zone is a cylinder.
 14. The method for theautonomous herding of flocks of birds of claim 11, wherein the protectedzone is over an airport.
 15. The method for the autonomous herding offlocks of birds of claim 11, wherein the at least one bird locationsensor is an avian radar.
 16. The method for the autonomous herding offlocks of birds of claim 11, further comprising obtaining, using theflock herding control system, environment data from at least oneenvironmental sensor describing environmental conditions proximal to theprotected zone from the at least one environmental sensor; andgenerating the set of waypoints using the environment data.
 17. Themethod for the autonomous herding of flocks of birds of claim 16,wherein the environment data describes the position of airplanes. 18.The method for the autonomous herding of flocks of birds of claim 16,wherein the environment data describes wind speed.
 19. The method forthe autonomous herding of flocks of birds of claim 16, furthercomprising: obtaining, using the flock herding control system, updatedbird position data; updating, using the flock herding control system,the set of waypoints based on the updated bird position data; uniformlysampling the updated set of waypoints to produce an updated subsample ofwaypoints; scoring each specific waypoint in the updated subsample ofwaypoints based on whether or not placing the unmanned aerial vehicle atthe specific waypoint will move the flock of birds in a desireddirection and the distance to other waypoints in the updated subsampleof waypoints; sorting the updated subsample of waypoints based on score;and selecting a new waypoint from the sorted, updated subsample ofwaypoints having the highest score.
 20. The method for the autonomousherding of flocks of birds of claim 16, further comprising: generating,using the flock herding control system, a second set of waypoints usingthe predictive flock dynamics model; uniformly sampling the second setof waypoints to produce a second subsample of waypoints; scoring eachspecific waypoint in the second subsample of waypoints based on whetheror not placing the unmanned aerial vehicle at the specific waypoint willmove the flock of birds in a desired direction and the distance to otherwaypoints in the second subsample of waypoints; sorting the secondsubsample of waypoints based on score; selecting a second waypoint fromthe sorted second subsample of waypoints having the highest score; andinstructing, using the flock herding control system, a second unmannedaerial vehicle to navigate to the selected second waypoint such that theflock of birds will, in response to the presence of the second unmannedaerial vehicle at the given at least one waypoint, change trajectoryaway from the protected zone.